Dedekind-Hasse norm: Difference between revisions
(New page: ==Statement== A '''Dedekind-Hasse norm''' on a commutative unital ring <math>R</math> is a function <math>N</math> from the nonzero elements of <math>R</math> to the set of nonnegativ...) |
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* <math>a</math> is an element of the ideal <math>(b)</math>. In other words, <math>b | a</math>. | * <math>a</math> is an element of the ideal <math>(b)</math>. In other words, <math>b | a</math>. | ||
* There is a nonzero element in the ideal <math>(a,b)</math> whose norm is strictly smaller than that of <math>b</math>. | * There is a nonzero element in the ideal <math>(a,b)</math> whose norm is strictly smaller than that of <math>b</math>. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Weaker than::Euclidean norm]]: {{proofofstrictimplicationat|[[Euclidean implies Dedekind-Hasse]]|[[Dedekind-Hasse not implies Euclidean]]}} | |||
* [[Weaker than::Multiplicative Dedekind-Hasse norm]] | |||
* [[Weaker than::Multiplicative Euclidean norm]] | |||
==Facts== | ==Facts== | ||
* A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat|[[Dedekind-Hasse norm implies principal ideal ring]]}} | * A commutative unital ring that admits a Dedekind-Hasse norm is a [[principal ideal ring]]. {{proofat|[[Dedekind-Hasse norm implies principal ideal ring]]}} | ||
Latest revision as of 18:24, 23 January 2009
Statement
A Dedekind-Hasse norm on a commutative unital ring is a function from the nonzero elements of to the set of nonnegative integers, satisfying the following condition:
Whenever are both nonzero, then one of these cases holds:
- is an element of the ideal . In other words, .
- There is a nonzero element in the ideal whose norm is strictly smaller than that of .
Relation with other properties
Stronger properties
- Euclidean norm: For proof of the implication, refer Euclidean implies Dedekind-Hasse and for proof of its strictness (i.e. the reverse implication being false) refer Dedekind-Hasse not implies Euclidean
- Multiplicative Dedekind-Hasse norm
- Multiplicative Euclidean norm
Facts
- A commutative unital ring that admits a Dedekind-Hasse norm is a principal ideal ring. For full proof, refer: Dedekind-Hasse norm implies principal ideal ring